The fact that

Ramanujan’s Constanteπ√163 is almost an integer (262537412640768743.99999999999925…) doesn’t seem to be a coincidence, but has to do with the 163 appearing in it. Can you explain why it’s almost-but-not-quite an integer in layman’s terms (I’m not a mathematician)?

**Answer**

This is quite a challenge to express in “layman’s terms”, but the

reason is that

j(1+√−1632)

is an integer where j is the j-function. When you substitute

(1+√−163)/2 into the q-expansion (see the wikipedia page)

of j, all terms save the first two are small, and the first two equal

−exp(π√163)+744.

The reason that this j-value is an integer is due to the quadratic

field Q(√−163) having

class number one, or equivalently that all positive-definite

integer binary quadratic forms of discriminant −163 are equivalent.

**Added**

I’ll try to explain the connection with binary quadratic forms. Consider

a quadratic form

Q(x,y)=ax2+bxy+cy2

with a, b and c integers. I’ll only consider forms Q which are

*primitive*, so that a, b and c have no common factor >1,

and *positive-definite*, that is a>0 and the *discriminant*

D=b2−4ac<0. There is a notion of equivalence of quadratic forms,

and two primitive positive-definite forms Q and Q′(x,y)=a′x2+b′xy+c′y2

(necessarily also of discriminant D) are equivalent if and only if

j(b+√−D2a)=j(b′+√−D2a′).

For each possible discriminant there are only finitely many equivalence

classes. Thus we get a finite set of j-values for each discriminant, and

the big theorem is that they are the solutions of a monic algebraic equation

with integer coefficients. When there is only one class the equation has the

form x−k=0 where x is an integer, and the j-value must be an integer.

My recommended reference for this is David Cox's book

Primes of the form x2+ny2. But these results appear towards

the end of this 350-page book.

**Attribution***Source : Link , Question Author : stevenvh , Answer Author : Robin Chapman*